AMonetaryTheory with Non-Degenerate Distributions

Transcription

1 AMonetaryTheory with Non-Degenerate Distributions Guido Menzio University of Pennsylvania Shouyong Shi University of Toronto This version: June 2013 Hongfei Sun Queen s University (hfsun@econ.queensu.ca) Abstract We construct and analyze a tractable search model of money with a non-degenerate distribution of money holdings. We assume search to be directed in the sense that buyers know the terms of trade before visiting particular sellers. Directed search makes the monetary steady state block recursive in the sense that individuals policy functions, value functions and the market tightness function are all independent of the distribution of individuals over money balances, although the distribution affects the aggregate activity by itself. Block recursivity enables us to characterize the equilibrium analytically. By adapting lattice-theoretic techniques, we characterize individuals policy and value functions, and show that these functions satisfy the standard conditions of optimization. We prove that a unique monetary steady state exists. Moreover, we provide conditions under which the steady-state distribution of buyers over money balances is non-degenerate and analyze the properties of this distribution. JEL classifications: E00, E4, C6 Keywords: Money; Distribution; Search; Lattice-Theoretic. Menzio: Department of Economics, University of Pennsylvania, 3718 Locust Walk Philadelphia, Pennsylvania 19104, USA. Shi: Department of Economics, University of Toronto, 150 St. George Street, Toronto, Ontario, Canada, M5S 3G7; Sun: Department of Economics, Queen s University, 94 University Ave., Kingston, Ontario, Canada, K7L 3N6. An associate editor and three referees have given comments that helped revising the paper. We have also received helpful comments from participants of seminars and conferences at the Baumol-Tobin conference at New York University (2012), the Canadian Macro Study Group meeting (Vancouver, 2011), the Society for Economic Dynamics (Istanbul, 2009 and Montreal, 2010), U. of Calgary (2010), Tsinghua Macro Workshop (Beijing, 2010), Singapore Management U. (2010), National Taiwan U. (2010), Chicago FED Conference on Money, Banking, Payments and Finance (Chicago, 2009 and 2010), Research and Money and Markets (Toronto, 2009), the Society for the Advancement of Economic Theory (Ischia, Italy, 2009) and the Texas Monetary Conference (Austin, 2009). Shi and Sun gratefully acknowledge financial support from the Social Sciences and Humanities Research Council of Canada. Shi also acknowledges financial support from the Bank of Canada Fellowship and the Canada Research Chair. The opinion expressed in the paper is our own and does not reflect the view of the Bank of Canada. All errors are ours.

2 1. Introduction Monetary search theory originated in Kiyotaki and Wright (1989) provides a strong microfoundation of money by deriving positive value of fiat money from underlying frictions in decentralized exchange, such as the lack of double coincidence of wants and anonymity of individuals. The theory has undergone significant development and has been shown to be useful for addressing a wide range of monetary issues in theory and policy. Despite all the development, monetary search theory has had only limited success in exploring a natural outcome of decentralized exchange a persistent, non-degenerate distribution of money balances. This distribution of money is important for both theory and policy. In this paper, we construct a tractable search model where a non-degenerate distribution of money can be persistent. We prove the existence of a unique monetary steady state, provide the conditions under which the distribution of money is non-degenerate, and examine the properties of the distribution. Decentralized exchange naturally induces a non-degenerate distribution of money. Because matching is stochastic, different individuals may end up trading away different amounts of money. Even if all individuals hold the same amount of money initially, the distribution of buyers over money balances can fan out as decentralized exchange continues. In the first generation of search models (e.g., Kiyotaki and Wright, 1989) and the second generation (e.g., Shi, 1995, and Trejos and Wright, 1995), the distribution of money is made degenerate by the assumption that an individual can hold either zero or one unit of indivisible money. Search models in the third generation allow both money and goods to be divisible, but they impose other assumptions to make money distribution degenerate. Notably, Shi (1997) assumes that each household consists of a large number of members who share consumption and utility, and so all households hold the same amount of money in the equilibrium. Lagos and Wright (2005) assume that each decentralized market is followed by a centralized market in which individuals have quasi-linear preferences over a good, and so trading in the centralized market eliminates all dispersion in money holdings. Green and Zhou (1998, GZ henceforth) are the first to formally analyze a non-degenerate money distribution, but they assume that goods come as a fixed endowment and money can only be accumulated in discrete units. Without these restrictive assumptions, an analytically tractable microfoundation of money with a non-degenerate distribution has eluded monetary theory. Very little is known about even the fundamental properties of an equilibrium. For example, does such a steady state exist, is it unique, and when is the distribution non-degenerate? A persistent, non-degenerate distribution of money is also important for policy analysis. One issue is the wealth effect of monetary policy. When individuals money balances differ, so do 1

3 their marginal values of money. A change in monetary policy has a redistributive effect that can affect aggregate welfare. A related issue is the effect of monetary policy on real aggregate activities. As documented by Christiano et al. (1999) using vector autoregression (VAR), even an impulse monetary shock can have a persistent effect on aggregate real activity. This persistence seems difficult to be captured by models where the distribution of money is degenerate or not persistent, such as Lucas (1990). In contrast, Rotemberg (1984) follows Baumol (1952) and Tobin (1956) to model an individual s money holdings as the solution to a problem of optimal inventory management. Because cash withdrawals are staggered across individuals, the distribution of money is non-degenerate in the steady state, and a monetary shock affects aggregate real activity persistently. These papers, and the large literature spawned from them, are useful indications of the importance of money dispersion in analyzing policy, but they assume an exogenous role of money rather than deriving this role from the fundamentals of the model. 1 We focus on the characterization of the steady state of a search economy with a non-degenerate distribution, although the broad research project is also motivated by the dynamic effect of monetary policy. This focus is a useful first step because it addresses the fundamental questions raised above on the equilibrium and provides a long-run anchor for a dynamic analysis. As a standalone contribution, the analysis provides a set of theoretical tools that help overcome the main difficulties in building a monetary theory with a non-degenerate distribution. The main deviation from the literature of monetary search that makes our model tractable lies in the way we model search. The literature assumes search to be undirected in the sense that individuals do not know the terms of trade before they are matched. In contrast, we assume search to be directed in the sense that individuals know the terms of trade before a match, as in Peters (1991), Moen (1997), Acemoglu and Shimer (1999), Burdett et al. (2001) and Julien et al. (2000). Directed search is a realistic feature of an actual economy. It reflects the fact that individuals have information about the location and the price range of the goods they want to buy. They go directly to the sellers who sell the goods they want, rather than randomly search among all the sellers. In our model, each type of good can be sold in many submarkets that specify different terms of trade and tightness (i.e., the ratio of trading posts to buyers). Buyers choose which submarket to visit and firms choose how many trading posts to create in each 1 Shi (2003) extends the large-household framework in Shi (1997) to examine the persistence of a monetary shock with search frictions. He introduces long-term nominal bonds with a partial legal restriction that prohibits a fraction of the individuals such as government agents from accepting bonds as a means of payment for goods and services. A monetary shock has a more persistent real effect than in Lucas (1990) because the shock affects the composition of money and bonds used in the goods market in future periods. However, the persistence is not as strong as empirically documented, partly because the large-household framework makes the distribution of asset holdings across households degenerate. 2

4 submarket. There is a cost of creating a trading post for a period, and the number of trading posts in each submarket is determined by free entry. Once individuals are inside a submarket, a frictional matching process determines the matching probability for a trading post or a buyer as a function of the tightness of the submarket. In equilibrium, the tightness in each submarket is consistent with buyers search choice and firms creation of trading posts. Directed search induces buyers to sort into submarkets. Specifically, because the marginal value of money is lower to a buyer who has a relatively high money balance, such a buyer has a strong desire to spend a relatively large amount of money on consumption and to spend it sooner rather than later. To do so, the buyer chooses to enter a submarket where he has a relatively high matching probability to trade a relatively large amount of money for a large quantity of goods. Firms cater to this desire by creating a relatively large number of trading posts per buyer in this submarket. Because buyers with different money holdings choose not to mix with each other, a buyer s optimal choices depend on the buyer s own money balance and the tightness of the particular submarket he visits, but not on the distribution of individuals over money balances. Moreover, because each submarket is tailored to only one group of buyers with a particular money balance, the tightness of each submarket that ensures zero profit for a trading post does not depend on the distribution of money. Precisely, individuals policy functions, value functions and the market tightness function are all independent of the distribution in the steady state. This feature of the steady state is referred to as block recursivity. With block recursivity, the distribution of money ceases to be part of the state space in individuals decision problems. This overcomes the main roadblock of tractability that arises in the literature when the endogenous distribution of money affects individuals decisions. As a result, we can characterize an individual s policy and value functions solely as functions of the individual s own balance. Having done so for each balance separately, we can compute the flows of individuals across money balances to obtain the distribution. The analytical tractability of the model enables us to prove that a unique monetary steady state exists, to determine when the steady-state distribution of buyers over money balances is non-degenerate, and to analyze the properties of this distribution. In the steady state, the support of the distribution consists of a finite number of values of money balance, each of which is associated with one active submarket. Moreover, an individual goes through purchasing cycles. When the individual has no money, he works to obtain money and then becomes a buyer. Starting with a high balance, a buyer enters a submarket where he has a high matching probability, spends a large amount of money and obtains a large quantity of goods. For the next trade, the buyer will go into a submarket where his matching probability, 3

5 the required spending and the quantity of goods obtained in a trade are all lower. The buyer will continue this pattern until he depletes his balance, at which point he will work again. The unique monetary steady state unifies the literature by nesting two well-known classes of models as special cases. In one case, the distribution of money is degenerate in the steady state, which occurs when individuals are sufficiently impatient. In this case, all buyers hold the same amount of money and spend the entire amount in one trade, and so a purchasing cycle consists of only one purchase. This endogenous pattern resembles the one assumed in the models with indivisible money (e.g., Shi, 1995, and Trejos and Wright, 1995). However, the endogenously generated pattern has a very different policy effect from the exogenously assumed pattern. Namely, a one-time change in the money stock does not affect the real activity in the steady state in our model, but it does in Shi (1995), Trejos and Wright (1995) and GZ. The other case of the unique steady state features a non-degenerate distribution of buyers over money balances. This case arises when individuals are sufficiently patient, together with other conditions specified in Theorem 4.2. In this case, each buyer runs down money balance in a purchasing cycle through consecutive purchases, and the purchasing pattern is staggered across the buyers. Moreover, because the buyers who hold a high balance trade relatively quickly and exit from that balance, there are more buyers with low balances than with high balances, and so the density of the distribution in a purchasing cycle is a decreasing function of money balance. The purchasing cycle resembles the one in the inventory model of money by Baumol (1952) and Tobin (1956), but the latter authors model the role of money in a reduced form. The decreasing density of money distribution resembles that in GZ, but GZ restrict that goods come as a fixed endowment and money can only be accumulated in discrete units. Moreover, in contrast to both Baumol-Tobin and GZ, we allow individuals to choose among submarkets that differ in the terms of trade and the matching probability. As a realistic feature, this endogeneity should be important for how monetary policy affects the aggregate activity. We will contrast our model further to the Baumol-Tobin model in subsection 4.1 and to GZ in subsection 4.3. A large part of this paper is devoted to the analysis of a buyer s decision problem, which establishes the properties of the policy and value functions. We provide a set of analytical tools to overcome some difficulties in the use of dynamic programming. The difficulties arise from the features that a buyer s objective function is not concave and that a buyer s value function cannot be assumed to be differentiable a priori. These difficulties prevent us from using the standard approach in dynamic programming (e.g., Stokey et al., 1989) to analyze the policy and value functions. In subsection 3.2.1, we will give an overview of these difficulties and the way in which we resolve them. A short description is that we adapt lattice-theoretic techniques (see 4

6 Topkis, 1998) to prove that a buyer s policy functions are monotone functions of the real balance. Using this result, we prove further that optimal choices obey the first-order conditions, the value functions are differentiable and the envelope conditions hold. By establishing these standard conditions formally, we hope to make the model easy to use. This procedure of analyzing a dynamic programming problem is of independent interest because it is applicable in a variety of dynamic models that involve both discrete and continuous choices. 2 It is important to clarify that our analysis and the main results do not follow simply from the labor search literature, despite the fact that this literature has explored directed search and block recursivity (e.g., Shi, 2009, Gonzalez and Shi, 2010, and Menzio and Shi, 2011). Several elements in our model are important for monetary theory, but not necessarily so for labor theory. First, an individual s gain from a monetary trade depends not only on how the match surplus is split, but also on how all individuals in the economy value money. A monetary equilibrium must determine this value of money. This is not necessary in a labor search model because the value of a match is determined by preferences there. Second, money balance is a stock variable that can be accumulated or decumulated over time through trade, and there is a market clearing condition on the aggregate stock of money. Such a stock variable has been absent in many labor-search models. Third, a buyer in a monetary model optimally chooses the length of a purchasing cycle and the amount of money to be spent in each period within a cycle. Most parts of this paper (e.g., sections 3 and 4) are devoted to resolving these monetary issues. Even at the technical level of using lattice-theoretic techniques, our analysis differs from that in Gonzalez and Shi (2010). While Gonzalez and Shi explore convexity of the value function to apply lattice-theoretic techniques, a buyer s value function in our model is neither convex nor concave. To apply lattice-theoretic techniques, we analyze a buyer s decision problem in steps (see subsection 3.2.2). In the literature of monetary search, Corbae et al. (2003) seem the first to incorporate directed search. They focus on the formation of trading coalitions and assume that money and goods are indivisible. Rocheteau and Wright (2005) examine directed search as a robustness check, Lagos and Rocheteau (2005) compare directed search with bargaining in the welfare effect of inflation, and Galenianos and Kircher (2008) and Julien et al. (2008) examine directed search with auctions. These papers do not formulate a block recursive equilibrium. Moreover, they follow Lagos and Wright (2005) to assume quasi-linear preferences that make either money distribution degenerate or its real effect temporary. 3 In addition, Lagos and Rocheteau (2005) apply lattice-theoretic 2 There are a large number of applications of lattice-theoretic techniques in dynamic programming and, specifically, in models of economic growth. Gonzalez and Shi (2010) use these techniques in a labor search model and provide a partial list of references. 3 Berentsen, Camera and Waller (2005) extend the Lagos-Wright (2005) model to allow for two rounds of trading 5

7 techniques to examine comparative statics. However, their assumption of quasi-linear preferences implies that the future value function is linear in money balance. This feature sidesteps the analytical difficulties overcome here that are intrinsic to search models where money has a wealth effect (see subsection 3.2.1). With undirected search, some papers have studied a non-degenerate money distribution. GZ are the first along this line, but they restrict that goods come as a fixed endowment and money can only be accumulated in discrete units. Zhou (1999) extends the GZ model by introducing a production cost. Berentsen et al. (2004) introduce lotteries into GZ. Zhu (2005) extends the GZ model by making goods divisible. He studies a sequence of economies with discrete money holdings and characterizes the limit where the size of the discreteness goes to zero. Eliminating the indivisibility restrictions, Molico (2006) and Chiu and Molico (2008) numerically compute the equilibrium. In particular, Chiu and Molico (2008) generate a non-degenerate money distribution by extending the Lagos-Wright model to allow the cost function in the centralized market to be strictly convex. These numerical exercises are useful, but they do not address the fundamental issues about an equilibrium with a non-degenerate money distribution that have eluded the literature, such as existence and uniqueness of the steady state. These models are analytically intractable precisely because the assumption of undirected search makes the equilibrium not block recursive. Even for a quantitative analysis, our model is easier to compute than these models of undirected search, as we will discuss in section 6. Moreover, undirected search models, including Molico (2006) and Chiu and Molico (2008), have several implications in contrast with our model, which we will discuss at the end of subsection A Monetary Economy with Directed Search 2.1. The model environment There are types of individuals and types of perishable goods indexed by {1 2 }, where 3. Each type consists of a continuum of individuals with measure one who are specialized in the consumption of good and the production of good + 1 (modulo ). The preferences of a type individual are represented by the utility function P =0 [ ( ) ( )], where (0 1) is the discount factor, : R + R is the utility of consumption of good, and :[0 1] R is the disutility of labor. We assume that is strictly increasing, strictly concave and twice in the decentralized market before a centralized market opens for a homogeneous good over which individuals have quasi-linear preferences. The non-degenerate distribution of money is not persistent, because it is re-set frequently by the trading in the centralized market. An extension of the Lagos-Wright model to allow for many consecutive rounds of decentralized trading is not tractable. Faig (2008) introduces lotteries into the Lagos-Wright model to partially avoid the use of quasi-linear preferences. Again, the money distribution is degenerate in his model. 6

8 continuously differentiable, with the boundary properties: (0) = 0, 0 ( ) = 0, and 0 (0) is sufficiently large. Similarly, we assume that is strictly increasing, strictly convex and twice continuously differentiable, with the boundary properties: (0) = 0 and 0 (1) =. In addition to consumption goods, there is an object called fiat money which is intrinsically worthless, perfectly divisible and costlessly storable. The supply of fiat money per capita is. We assume that is constant in most parts of this paper and will allow for money growth in section 5. The economy is also populated by types of firms. Each type consists of a large number of firms that are specialized in the production and distribution of good. Atype firm operates a technology of constant returns to scale that transforms any amount of labor supplied by individuals of type 1 (modulo ) intothatamountofgood. 4 Moreover, a type firmcanopena trading post in the market for good using 0 units of labor supplied by individuals of type 1 (modulo ). Firms are owned by the individuals through a balanced mutual fund. In every period, a labor market and a product market open. Firms can participate in both markets in the same period. In contrast, individuals can participate in either the labor market or the product market. That is, in a given period, individuals must choose whether to become workers or buyers. Before making this choice, individuals can play a fair lottery. Even though individuals are risk averse, a lottery can be desirable because the value function without the lottery can be non-concave at particular money balances. One cause of non-concavity is the discrete nature of the decision on which market to enter. Another cause is the tradeoff between the matching probability and the surplus of trade in the product market, to be described later. The labor market is centralized and frictionless. Taking the nominal wage rate as given, each firm chooses how much labor to demand and each worker chooses how much labor to supply. In equilibrium, the nominal wage rate equates the demand for and the supply of labor of each type. All labor is paid at the end of the period from the proceeds of sales. Workers are paid in money instead of goods because they do not want to consume the good produced by the firm in which they work and because goods are perishable between periods. Moreover, an IOU issued by a firm that promises future repayment is not accepted, because the firm is better off exiting the market than honoring such IOUs. 5 4 The linear production technology is assumed without loss of generality. Note that the disutility of labor supply, ( ), is strictly convex and the utility of consumption is strictly concave. 5 Firms are assumed to commit to the wage payment at the end of the period. Similar assumptions of withinperiod commitments are required in all models of decentralized exchange where goods are perishable, such as Shi (1995) and Trejos and Wright (1995). In the latter models, the two traders in a match are assumed to commit to the delivery of money and goods according to the bargained terms of trade. Without this assumption, a buyer would renege and ask the seller to give up the goods for free once the goods are produced. Although a detailed description of the environment can be given to justify the within-period commitment, it is often skipped in the literature, as it is here, because it would take the attention away from the main issues. 7

9 To simplify the notation, we choose labor, instead of goods or money, as the numeraire in this model. Let be the nominal wage rate, and so one unit of money is worth 1 ( ) units of labor. We refer to the quantity of money expressed in terms of labor as the real balance. Thus, the real balance per capita in the economy is equal to 1. We will also express the price of goods in labor units later. Although is normalized by the money stock, we shorten the phrase by referring to as the nominal wage rate whenever there is no confusion. The product market is decentralized and has search frictions. Buyers and trading posts meet in pairs and there is no record keeping of their actions once they exit a trade. The market for each type goods is organized in a continuum of submarkets indexed by the terms of trade ( ) R + R +,where is the real balance paid by the buyer and is the quantity of goods obtained by the buyer in a trade. A firm chooses how many trading posts to create in each submarket, and a buyer chooses which submarket to visit. As is standard in search models, the length of a period is such that a buyer can visit at most one submarket in a period. The matching process is frictional in each submarket. Let denote the tightness, i.e., the ratio of trading posts to buyers, in a submarket. In a submarket with tightness, a buyer is matched with probability = ( ), and a trading post is matched with probability = ( ). The function : R + [0 1] is a strictly increasing function with boundary conditions (0) = 0 and ( ) = 1. The function : R + [0 1] is a strictly decreasing function such that ( ) = ( ), (0) = 1 and ( ) =0. Since and are both functions of, we can express a trading post s matching probability as a function of a buyer s matching probability: = ( ) ( 1 ( )). Clearly, ( ) is a decreasing function. We assume that 1 ( ) is strictly convex in. Because firms and buyers choose which submarket to enter, a type buyer will choose to participate only in the submarkets where type goods are produced, i.e., where trading posts are created by type firms. Moreover, across the submarkets that sell the same good, search is directed as in Moen (1997), Acemoglu and Shimer (1999), Burdett et al. (2001) and Shi (2001). That is, firms and buyers take into account the fact that market tightness varies with the terms of trade across the submarkets according to a function : R + R + R +. The function is endogenously determined in the equilibrium by the requirement that ( ) beequaltotheratio of trading posts to buyers in submarket ( ) for all ( ). As a result, matching probabilities, and, are endogenous functions of ( ). 6 When a buyer meets a trading post in submarket ( ), the buyer pays the real balance for 6 Note that the price of goods in a submarket alone is not adequate for describing a submarket because a buyer may not spend all the money in a trade. Also, in subsection 2.4, we will briefly contrastdirectedsearchwitha perfectly competitive goods market and a trading-post model with no search frictions. 8

10 units of the consumption good. The buyer must pay the seller with money because neither barter nor credit is feasible. The buyer cannot pay the seller with goods because goods are perishable and there is no double coincidence of wants in goods between the buyer and the seller. Moreover, the buyer cannot pay the seller with an IOU because individuals are anonymous; once they exit a trade, they can renege on their IOUs without fear of retribution. Thus, the amount of money that a buyer can spend in a trade is bounded above by the balance he carries into the trade. The environment above retains many standard features of a search model. In particular, matching is stochastic in each submarket, an individual can perform only one role in a period, and a buyer has at most one match in a period. Directed search is the main difference of our environment from the literature of monetary search. In order to direct search, some commitment is needed for individuals to make a meaningful tradeoff between the surplus of a trade and the matching probability. In our paper, the commitment is to the posted terms of trade. Let us clarify this assumption with three remarks. First, directed search is a realistic feature of many markets in an actual economy. The trading posts in a submarket may be interpreted as a collection of stores in a particular location or sellers who offer similar terms of trade. Buyers often choose the location and the price range of goods at which they shop, rather than randomly search among all the sellers. Second, modeling a market as a collection of trading posts can be traced at least back to Shapley and Shubik (1977), who argue that the setup is useful for retaining the force of competition while allowing for coordination frictions. 7 Third, commitment to the terms of trade is a simple way to model directed search, but it is not the only one. Search can be directed by commitment to a variety of trading arrangements, from posted prices (e.g., Burdett et al., 2001) to auctions (Julien et al., 2000). Directed search has other desirable implications for modeling monetary exchange, which contrast with undirected search models (e.g., Shi, 1997, Lagos and Wright, 2005, Molico, 2006, and Chiu and Molico, 2008). First, undirected search models have often (although not always) assumed that the individuals in a match bargain over the terms of trade. To characterize the bargaining outcome simply, the models require a strong assumption that individuals money holdings are public information. If money holdings are private information, instead, the bargaining game can have multiple equilibria, which complicate the analysis significantly and weaken the predictive power. Our model does not suffer from this problem. Because search is directed, whether an individual s money balance is public or private information does not matter to the analysis. In 7 The main difference between directed search and the Shapley-Shubik setup is that with directed search, matching is frictional and stochastic inside each submarket, and so there is a trade-off between the terms of trade and the matching probability. In the Shapley-Shubik setup, all participants at a post succeed in trade, provided that the number of participants on each side of the post is strictly positive. 9

11 any submarket, the expected payoff to an individual is determined by the terms of trade and the individual s own money balance. An individual can calculate this expected payoff without the need to know how much money the trading partner holds. Second, with undirected search and bargaining (under public information), a buyer with a higher balance pays a higher price in the equilibrium not because the buyer optimally chooses so, ex ante, but rather because the buyer is held up by the seller, ex post. In our model, as will be proven later, a buyer with a higher balance optimally chooses to pay a higher price because the submarket with a higher price compensates the buyer with a higher matching probability An individual s decision Let ( ) denote the lifetime utility of an individual who starts a period with the real balance (expressed in units of labor). We refer to as the ex-ante value function, since it is measured before the individual makes any decision in a period. Let ( ) denoteabuyer s value function, i.e., the lifetime utility of an individual who enters the product market as a buyer with the real balance. Similarly, let ( ) denoteaworker s value function, i.e., the lifetime utility of an individual who enters the labor market as a worker with the real balance. A worker chooses labor supply,, which generates units of the real balance as wage income. The individual also owns a diversified portfolio of the firms. However, the return to this portfolio is zero since all firms earn zero profit in the equilibrium. Thus, a worker who enters the labor market with a real balance will have a real balance + at the end of the period. The discounted value of this balance is ( + ). The worker s value function,,obeys: ( ) = max[ ( + ) ( )] (2.1) [0 1] Denote the optimal choice of as ( ) and the implied real balance at the end of the period as ( ) = + ( ). We refer to ( ) and ( ) as a worker s policy functions. A buyer chooses which submarket ( ) to enter, taking the tightness function ( ) as given. In submarket ( ), the buyer will meet a trading post with probability ( ( )), in whichcasehewilltradearealbalance for units of goods. Current consumption yields utility ( ), and the residual balance ( ) yields the discounted value ( ). With probability 1 ( ( )), the buyer will not have a match, in which case he will retain the balance whose discounted value will be ( ). Thus, the buyer s value function,, obeys: ( ) = max [0 ], 0 { ( ( )) [ ( )+ ( )] + [1 ( ( ))] ( )}. (2.2) 10

12 The buyer s optimal choices are represented by the policy functions ( ( ) ( )). An individual chooses whether to be a worker or a buyer in the period. This choice induces: ( ) =max{ ( ) ( )} (2.3) Notice that may be non-concave for some real balances, even when and are concave functions. Thus, there is a potential gain to the individual from playing fair lotteries before making the above choice on whether to be a worker or a buyer. Denote a lottery as ( ) =1 2, with 2 1,where is the probability that the prize is realized. The lifetime utility of the prize is ( ). Thus, the ex ante value function induced by the lottery choice is: h i ( ) = max 1 ( 1 )+ 2 ( 2 ) (2.4) ( ) s.t =, =1, 2 1, [0 1] and 0for =1 2 Let ( ( ) ( )) =1 2 denote the individual s optimal choice of a lottery A firm s decision A firm chooses how many trading posts to create in each submarket and how much labor to employ, taking as given the wage, 1, and the tightness function, ( ). The firm hires labor to create trading posts and produce goods, and pays labor with money received from selling goods. Consider submarket ( ). The cost of creating a trading post is units of labor. A trading post in submarket ( ) willhaveamatchwithprobability ( ( )), in which case the firm uses units of labor to produce units of goods and exchanges for a real balance. Thus, the expected benefit of creating a trading post in submarket ( ) is ( ( ))( ). If ( ( ))( ), it is optimal for the firm not to create any trading post in submarket ( ). If ( ( ))( ), it is optimal for the firm to create infinitely many trading posts in submarket ( ). If ( ( ))( ) =, thefirm is indifferent between creating different numbers of trading posts in submarket ( ). The case ( ( ))( ) never occurs, because it implies ( ) = and, hence, ( ( )) = 0, which contradicts the condition for the case. Thus, in any submarket ( ) visited by a positive number of buyers, the tightness is consistent with the firm s incentive to create trading posts if and only if ( ( ))( ) and ( ) 0, (2.5) 8 Because a lottery is defined for any given, it is used by individuals who hold the same balance. Thus, a lottery is not introduced here for individuals with heterogeneous holdings to exchange their balances among each other between trading rounds. Moreover, a lottery may or may not be played in the equilibrium and, when it is played in the equilibrium, it is played at only one level of money balance (see Lemma 4.1). 11

13 where the two inequalities hold with complementary slackness. In any submarket ( ) that is not visited by buyers, the tightness can be arbitrary if is greater than ( ( ))( ). However, following Shi (2009), Menzio and Shi (2010, 2011) and Gonzalez and Shi (2010), we restrict attention to equilibria in which (2.5) also holds for such submarkets. 9 Note that (2.5) implies that aggregate profit is zero. If ( ) is the distribution of trading posts across submarkets, then the sum of money received from sales, R ( ( )) ( ), exactly covers the sum of money paid to the workers, R [ + ( ( )) ] ( ). This implies that expected profit ofeachfirm is zero. Basing on this result, we assume that actual profit ofeach firm is also zero. To justify this assumption, one may imagine that the number of firms is finite and each firm creates a large number of trading posts so that the law of large numbers applies to each firm to guarantee deterministic revenue and cost for each firm. Although each firm in this case has some size in the market, it is not uncommon to assume firms as price takers in the labor market as we do. For example, the celebrated work of Debreu (1959) assumes that a finite number of firms and a finite number of households take prices parametrically, even though their actions affect equilibrium prices Equilibrium definition and block recursivity We define a monetary steady state as follows: Definition 2.1. A monetary steady state consists of value functions, ( ), policy functions, ( ), market tightness function, a wage rate, and a distribution of individuals over real balances,, that satisfy the following requirements: (i) satisfies (2.1) with as the associated policy function; (ii) satisfies (2.2) with ( ) as the associated policy functions; (iii) satisfies (2.4) with ( ) as the associated policy functions; (iv) satisfies (2.5) for all ( ) R 2 +; (v) is the ergodic distribution generated by ( ); 10 (vi) is such that and R ( ) =1. 9 This restriction on the beliefs out of the equilibrium completes the market in the following sense: A submarket is inactive only if, given that some buyers are present in the submarket, the expected benefit to a lone trading post in the submarket is still lower than the cost of the trading post. This restriction can be justified by a tremblinghand argument that a small measure of buyers appear in every submarket exogenously. Similar restrictions are common in the literature on directed search, e.g., Moen (1997) and Acemoglu and Shimer (1999). 10 The general specification of the law of motion of is cumbersome at this point and not necessary for the equilibrium analysis. In section 4 we will characterize the law of motion of implied by optimal choices. 12

14 Requirements (i)-(iv) are explained in previous subsections. Requirement (v) asks the distribution of individuals over real balances to be stationary and consistent with the flows of individuals induced by optimal choices. Requirement (vi) asks that money should have a positive value and that all money should be held by the individuals. Specifically, the sum of real balances according to the distribution must be equal to the total real balance in the economy, 1. We did not specify the labor market clearing condition in the above definition, because such a condition is implied by requirement (vi) in a closed economy. As we will show in section 4, the support of the equilibrium distribution is a discrete set. However, for individuals to optimally choose to hold only the balances in this set, they need to know the optimal choice and the payoff of holding any balance outside the equilibrium set. This information is provided by the policy and value functions in (i)-(iii) above. Similarly, although the number of submarkets participated by a positive measure of individuals is finite in the equilibrium, individuals need to know the tightness in all submarkets, given by the function in (iv) above, in order to choose optimally which submarket to participate in (see footnote 9). Equilibrium objects and requirements in Definition 2.1 can be grouped into two blocks. The first block consists of the value functions, the policy functions and the market tightness function, which are determined by requirements (i) - (iv). The second block consists of the distribution of individuals over real balances and the wage rate, which are determined by requirements (v) and (vi). The second block depends on the objects in the first block, but the first block is not affected by the second block. That is, the value functions, the policy functions and the market tightness function are independent of the distribution and the wage rate. We refer to this property of the equilibrium as block recursivity, a phrase coined by Shi (2009), Menzio and Shi ( 2010, 2011), and Gonzalez and Shi (2010). Clearly, even when an equilibrium is block recursive, the distribution isimportantbecauseitaffects the aggregate activity. Block recursivity is an attractive property of our model because it enables us to solve for equilibrium value functions, policy functions and the market tightness function without solving for the distribution of individuals over real balances. After obtaining the objects in the first block, we can compute the equilibrium distribution by simply equating the flows of individuals into and out of each level of the real balance. Thus, the steady state is tractable even when the distribution of real balances is non-degenerate. In contrast, when the distribution is an aggregate state variable that appears in the policy and value functions, one must compute the objects in the two blocks simultaneously and, since the distribution is endogenous and potentially has a large dimension, the computation of an equilibrium is complicated. In fact, it is to circumvent this complexity that monetary models have imposed assumptions on the model environment to 13

15 make the distribution degenerate (e.g., Shi, 1997, Lagos and Wright, 2005). Directed search and free-entry of trading posts are responsible for the steady state to be block recursive. With directed search, individuals choose to enter only the submarkets with the best tradeoff between the terms of trade and the matching probability, as formulated in subsection 2.2. The tightness function provides all information on the market that is relevant for this tradeoff. Given the tightness in each submarket, a buyer s optimal decision on which submarket to visit depends only on the buyer s own real balance, and not on how real balances are distributed among other buyers. Similarly, given the tightness in each submarket, the expected profit ofatrading post in a submarket depends only on the particular real balance of the buyers who are expected to enter that submarket, and not on how real balances are distributed in other submarkets. In turn, the tightness in each submarket is determined by free-entry of trading posts, which drives the expected profit ofatradingpostdowntozerowherever, and the tightness to zero wherever. Because the expected profit of a trading post in each submarket depends only on the real balance of the buyers who will enter that submarket, so does the resulting tightness. Thus, value functions, policy functions, and the market tightness function are all independent of money distribution in the steady state. To appreciate the role of directed search, consider an environment with undirected search in which all buyers and trading posts go through the same random matching process first and then decide whether to trade. The terms of trade can be either posted before the meeting (without serving the function of directing search) or bargained after the meeting. If the terms of trade are posted before a meeting takes place, whether a particular match generates a non-negative surplus depends on the real balance of the buyer in the match. Because the buyer is randomly drawn, the trading probability depends on the distribution of buyers over real balances. If the terms of trade are instead bargained after a meeting takes place, they depend on real balances of both individuals in the match which are randomly drawn from the distribution. In both cases, the distribution of individuals over real balances affects the value function and the expected benefit of a trading post. Because the tightness of the market is such that the expected benefit ofa trading post is equal to the cost, the tightness is also a function of the distribution. That is, when search is undirected, the equilibrium is not block recursive. Finally, let us remark on the assumption that the labor market is perfectly competitive. Although this assumption is standard in macro, it is not used in most money-search models which, instead, assume that a worker s income depends on the outcome of random matching (e.g., Shi, 1995, and Trejos and Wright, 1995). In our model, each firm hires workers to produce goods and to maintain a large number of trading posts. Although each trading post may or 14

16 may not have a trade, the law of large numbers implies that a firm s total revenue from all trading posts together is deterministic, which enables the firm to pay a deterministic competitive wage rate. Moreover, we will show in section 4 that all workers go to work with zero balance. However, neither the deterministic wage income nor the degenerate real balance among workers in the equilibrium is important for block recursivity. This should be clear from the fact that block recursivity is a statement about the independence of the entire value functions, policy functions and the tightness function on the distribution, not just the independence at particular real balances. For example, if there are idiosyncratic shocks to the disutility of labor or a worker s matching outcome, then the distribution of wage income among workers will be non-degenerate in the equilibrium. But the equilibrium will still be defined in the same way as above and will remain block recursive. We abstract from such heterogeneity among workers in order to focus on a buyer s decision and money distribution among the buyers. 3. Equilibrium Policy and Value Functions In this section we establish existence, uniqueness and other features of value and policy functions. A center piece of this analysis is subsection 3.2 on a buyer s value and policy functions. We prove that a buyer s policy functions are monotone, which implies that buyers sort into different submarkets according to the real balance. A buyer with a higher balance chooses to search in a submarket where he can spend a larger balance and get a higher quantity of goods. Such a buyer also has a higher matching probability. Sorting leads to a stylized pattern of purchases and a clear characterization of the equilibrium in section 4. Monotonicity of policy functions is also critical for us to prove that the standard conditions of optimization, such as the first-order conditions and the envelope conditions, hold in our model. The characterization of a buyer s problem is technically challenging because the problem is not well-behaved. In fact, a buyer s objective function is not concave in the choice and state variables jointly. We cannot use standard arguments in dynamic programming (e.g., Stokey et al., 1989) to establish monotonicity of the policy functions and differentiability of the value function and, in turn, to establish the validity of the envelope and first-order conditions. Instead, we develop an alternative set of arguments that first prove monotonicity of the policy functions, then differentiability of the value function and finally the validity of the first-order and envelope conditions. These arguments are of independent interest because they are likely to apply to a variety of dynamic models that involve both discrete and continuous choices. A map of the analysis in this section is as follows. First, we assume that an individual s real 15

17 balance is bounded above by, which will be validated in Theorem 3.5. Let C[0 ] denote the set of continuous and increasing functions on [0 ], and let V[0 ] denote the subset of C[0 ] that contains all concave functions. Taking an arbitrary ex ante value function V[0 ], we use subsection 3.1 to characterize a worker s problem. Second, with the same function,weuse subsection 3.2 to characterize a buyer s problem. Third, in subsection 3.3, we characterize an individual s lottery choice and obtain an update of the ex ante value function, denoted as. We prove that is a monotone contraction mapping on V[0 ], and so there is a unique fixed point for the ex ante value function. Finally, we verify in Theorem 3.5 that an individual s real balance is indeed bounded above by A worker s value and policy functions Let be a sufficiently large upper bound on individuals real balances and any arbitrary function in V[0 ]. Given, the worker s problem, (2.1), generates the worker s value function ( ), the policy function of labor supply ( ), and the policy function of the end-of-period balance ( ) = + ( ). We have the following lemma (see Appendix A for a proof): Lemma 3.1. For any [0 ] and V[0 ], the following properties hold: (i) V[0 ], i.e., is continuous, increasing and concave on [0 ]; (ii) ( ) and ( ) are single-valued and continuous, with ( ) being decreasing and ( ) strictly increasing; (iii) For all such that ( ) 0, 0 ( ) and 0 ( ( )) exist and satisfy: 0 ( ) = 0 ( + ( )) = 0 ( ( )). (3.1) The first equality is the envelope condition and the second equality the first-order condition. A worker s value function is continuous, increasing and concave in the worker s real balance because the ex ante value function has these properties. Optimal labor supply is decreasing in the worker s balance because the marginal value of money is diminishing and the marginal disutility of labor is increasing. The end-of-period balance is strictly increasing in because money has a strictly positive value. To establish differentiability of in (iii), we use the standard approach in dynamic programming (see Stokey et al., 1989, p85). That is, we first show that the objective function in (2.1) is concave in ( ) jointly, and then use the result in Benveniste and Scheinkman (1979) to show that is differentiable whenever the optimal choice ( ) is interior The choice = 1 is never optimal, because the marginal disutility of labor at this choice is infinite. On the other hand, if is so high that ( ) = 0, then the individual should have entered the goods market as a buyer rather than the labor market as a worker. 16